introduction to dynamical systemsintroduction to dynamical systems herbert wiklicky...

Introduction toDynamical Systems

Herbert [email protected]

Imperial College London

Bertinoro, June 2013

Herbert Wiklicky Dynamical Systems

The Land of Oz

The Land of Oz is blessed with many things, but not by goodweather. They never have two nice days in a row. If they have anice day, the chance of rain or snow the next day are the same.If there is rain or snow the chances are even that the weatherstays the same for the next day. If there is a change from snowor rain, only half of the time is this a change to a nice day.

The Land of Oz

From this we obtain the transition probabilities between nice(N), rainy (R) and snowy (S) days:

2 8814

S 12ff

The Land of Oz

We can then define the following transition matrix:

12 0 1

From Grinstead & Snell: Introduction to Probability, p406;available as GNU book on http://www.dartmouth.edu/∼chance

The Land of Oz

We can then define the following transition matrix:

12 0 1

From Grinstead & Snell: Introduction to Probability, p406;available as GNU book on http://www.dartmouth.edu/∼chance

Discrete Time Markov Chain

Given a finite set of states S = s1, . . . , sr.

A discrete time Markov chain (DTMC) on S is defined via astochastic matrix P as a above, i.e. an r × r (square) matrixwith entries 0 ≤ pij ≤ 1 and such that all row sums are equal toone, i.e. ∑

pij = 1.

Discrete Time Markov Processes

Let P be the transition matrix of a DTMC. The entry in p(n)ij in

the n-th matrix power Pn gives the probability that the Markovchain, starting in state si , will be in state sj after exactly n steps.

At any time step we can describe the probabilities of being in acertain state si by a probability ui . These probabilities define aprobability distribution, i.e. a row vector

u = (u1,u2, · · · ,ur )

such that 0 ≤ ui ≤ 1 and∑

ui = 1.

For any stochastic matrix P and probability distribution u themultiplication uP is again a probability distribution.

u = (u1,u2, · · · ,ur )

ui = 1.

u = (u1,u2, · · · ,ur )

ui = 1.

The Land of Oz

Consider the initial probability distributions u = (0,1,0) andv = (1

3 ,13 ,

13) in the Oz Example. The vector u describes a

situation where we are certain that we start with a nice day (N),while v corresponds to one where we assume the samechances of having a rainy (R), nice (N) or snowy (S) day.

The Land of Oz

3 ,13 ,

13) in the Oz Example.

(12,0,

)uP2 =

(38,14,38

The Land of Oz

3 ,13 ,

13) in the Oz Example.

0.416670.166670.41667

0.395830.208330.39583

0.401040.197920.40104

0.399740.200520.39974

· · · vP100 =

0.400000.200000.40000

Convention

Note that in the theory of Markov chains one usually isconcerned with probability distributions as row vectors.Therefore, probability vectors are post-multiplied by thestochastic matrix P defining a Markov chain.

The usual pre-multiplication could be realised via:

Pu = (uT PT )T

Dynamical Systems

Dynamical Systems (Birkhoff 1927)

Introductory remarks. In dynamics we deal with physicalsystems whoes state at time t is completely specified by thevalues of n real variables

x1, x2, . . . , xn.

Accordingly the system is such that the rates of change ofthese variables, namely

dx1/dt ,dx2/dt , . . . ,dxn/dt ,

merely depend upon the values of the variables themselves, sothat the laws of motion can be expresses by means of ndifferential equations of the first order

dxi/dt = Xi(x1, x2, . . . , xn) (i = 1, . . . ,n).

George D. Birkhoff. Dynamical Systems, volume 9 ofColloquium Publications. AMS, 1927.

Dynamical Systems (Bhatia/Szegö 1970)

. . . the symbol X denotes a metric space [. . . ] and R stands forthe set of real numbers.

1.1 Definition. A dynamical system on X is a triplet (X ,R, π),where π is a map from the product space X × R into the spaceX satisfying the following axioms:

1.1.1 π(x ,0) = x for every x ∈ X (identity axiom),1.1.2 π(π(x , t1), t2) = π(x , t1 + t2) for every x ∈ X and

t1, t2 ∈ R (group axiom),1.1.3 π is continuous (continuity axiom).

Nam Parshad Bhatia and Giorgio P. Szegö. Stability Theory ofDynamical Systems, volume 161 of Grundlehren dermathematischen Wissenschaften.

Dynamical Systems

DefinitionA general dynamical system is a triple (G, π,X ) with (G, ·) agroup, X any set and π : G × X → X with:Identity Axiom

π(e, x) = x

for all x ∈ X and e ∈ G unit.Homomorphism Axiom

π(g, π(h, x)) = π(gh, x)

for all x ∈ X and g,h ∈ G.

Elements of a General Dynamical System

A general dynamical systems is made up of three ingredients:

Phase Space: a set X where “things happen”. This can haveadditional structure (topology, norm, etc.)

Phase Group: the group G which allows us to “combine” thepartial dynamics to obtain a global picture.

Group Action: the way in which the dynamics of the group G isimplement on the phase space X .

Variations of Dynamical System

Typical phase groups are Z (integers) or R (reals) for so calleddiscrete time or continuous time models.

To investigate, for example, symmetries of the phase space it isalso often the case that one considers so-called Lie Groups astransformation groups.

Typically we will request that the group action preserves thestructure of the phase space, i.e. π(g, .) is a structurepreserving morphism on X for all g ∈ G.

An option is to drop invertability to get one-sided dynamicalsystems by taking G to be a semi-group (e.g. the naturals N).

(Phase) Groups

DefinitionA group G is a set with two maps (product and inverse)

. · . : G ×G→ G and .−1 : G→ G

fulfilling:

(i) (xy)z = x(yz) for all x , y , z ∈ G associativityaxiom.

(ii) ∃e ∈ G such that ex = xe = x f or all x ∈ Gidentity axiom.

(iii) x−1x = xx−1 = e for all x ∈ G inverse axiom.

Here the group is presented multiplicatively, some groups arerepresented additively, e.g. (Z,+) and (R,+).

(Phase) Groups

DefinitionA group G is a set with two maps (product and inverse)

. · . : G ×G→ G and .−1 : G→ G

fulfilling:

(i) (xy)z = x(yz) for all x , y , z ∈ G associativityaxiom.

(ii) ∃e ∈ G such that ex = xe = x f or all x ∈ Gidentity axiom.

(iii) x−1x = xx−1 = e for all x ∈ G inverse axiom.

Here the group is presented multiplicatively, some groups arerepresented additively, e.g. (Z,+) and (R,+).

G-Spaces

DefinitionLet G be a group. A G-Space is a set S and a mapτ : G × S → S so that

τ(e, s) = s all s ∈ S

andτ(g, τ(h, s)) = τ(gh, s)

for all g,h ∈ G and s ∈ S. τ is also called an action of G on S.

We write τg(s) = τ(g, s) so τg : S → S and we have τgτh = τghas well as τgτg−1 = τg−1τg = τe = id, see e.g. [3, I.2]

Phase Spaces

There are many choices for the phase space of a dynamicalsystem, among them we could mention:

Topological Spaces and require that π(g, .) a homeomorphism.Measurable Spaces with π(g, .) to be measure preserving.Vector Spaces like Rn with π a linear map or operator.Strings of Symbols in an alphabet Σ as in Symbolic Dynamics.Differentiable Manifolds as, e.g., in Classical Mechanics.Topological Vector Spaces, Toplogical Groups, Lie Groups, . . .

Phase Spaces

Group Action

DefinitionLet (G, π,X ) be a dynamical system. The orbit of a point x ∈ Xis given by

OG(x) = π(g, x) | g ∈ G.

DefinitionLet (G, π,X ) be a dynamical system. The group action π is

transitive iff∀x , x ′ ∈ X : OG(x) = OG(x ′).

faithful iffg 7→ π(g, x) is injective.

free iff∀x ∈ X : g 7→ π(g, x) is injective.

Group Action

OG(x) = π(g, x) | g ∈ G.

Group Action

OG(x) = π(g, x) | g ∈ G.

Group Action

OG(x) = π(g, x) | g ∈ G.

Group Action

OG(x) = π(g, x) | g ∈ G.

Elements of Ergodic Theory

Topological Dynamical System

DefinitionA topological dynamical system is a dynamical system (G, π,X )with the elements:

G is a topological group, i.e. . · . is continuous,X is a topological space,

and π fulfills theContinuity Axiom:

π : G × X → X is continuous.

Toplological Spaces

DefinitionA topological space is a set X together with a family of sub-setsτ ⊆ P(X ), the topology (of open sets), iff

1 ∅ ∈ τ and X ∈ τ .

n⋂i=0

Oi ∈ τ for Oi ∈ τ (finite).

3⋃i∈I

Oi ∈ τ for Oi ∈ τ (arbritrary).

The sets O ∈ τ are called open sets. The complementsA = X \O of open sets are closed sets.

Metric Spaces

DefinitionA metric space is a set X and a real-valued function d(., .), ametric, on X × X which satisfies:

1 d(x , y) ≥ 02 d(x , y) = 0 ⇐⇒ x = y3 d(x , y) = d(y , x)

4 d(x , z) ≤ d(x , y) + d(y , z)

Complete Metric Spaces

In a metric space we can define a basis for the topology opensets via open balls, i.e. sets B(x , ε) = x ′ | d(x , x ′) < ε, i.e.open sets are those which are unions of open balls.

Given a sequence (xi)i∈N of points in a topological space. Wesay that it converges if there exists x = lim xi such that for allneighbourhoods U(x) of x there ∃N s.t. for n > N : xn ∈ U(x).

A sequence of elements (xi)i∈N in a metric space (X ,d) iscalled a Cauchy sequence if

∀ε > 0 ∃N : n,m ≥ N ⇒ d(xn, xm) < ε.

A metric space (X ,d) in which all Cauchy sequences convergeis called complete (metric) space.

∀ε > 0 ∃N : n,m ≥ N ⇒ d(xn, xm) < ε.

Continuous Functions

DefinitionA function T : X → X ′ between two topological spaces (X , τ)and (X ′, τ ′) is calledcontinuous iff

∀O ∈ τ ′ : T−1(O) ∈ τ.

homeomorph iff

T is a bijection, and T and T−1 are continuous.

Continuous functions preserve limits, i.e. lim T(xi) = T(lim(xi)).

∀O ∈ τ ′ : T−1(O) ∈ τ.

homeomorph iff

∀O ∈ τ ′ : T−1(O) ∈ τ.

homeomorph iff

∀O ∈ τ ′ : T−1(O) ∈ τ.

homeomorph iff

Measure Theoretical Dynamical System

DefinitionA measure theoretic dynamical system is a dynamical system(G, π,X ) with

G is a measur(abl)e space,X is a measur(abl)e space,

and π fulfills theMeasurability Axiom:

π : G × X → X is measurable.π(g, .) : X → X is measure preserving ∀g ∈ G.

One can define a product measure on G × X in order to makesense of the first condition.

Measure Theoretical Dynamical System

DefinitionA measure theoretic dynamical system is a dynamical system(G, π,X ) with

G is a measur(abl)e space,X is a measur(abl)e space,

and π fulfills theMeasurability Axiom:

π : G × X → X is measurable.π(g, .) : X → X is measure preserving ∀g ∈ G.

One can define a product measure on G × X in order to makesense of the first condition.

Measureable Spaces

DefinitionGiven any set X . A family σ of sub-sets σ ⊆ P(X ) is called aσ-algebra iff

1 ∅ ∈ σ and X ∈ σ.

∞⋂i=0

Si ∈ σ for Si ∈ σ (countable).

3 X \ S ∈ σ for S ∈ σ.We say that (X , σ) is a measurable space, and S ∈ σ aremeasurable sets.

By de Morgan we have also:∞⋃

Measureable Spaces

DefinitionGiven any set X . A family σ of sub-sets σ ⊆ P(X ) is called aσ-algebra iff

1 ∅ ∈ σ and X ∈ σ.

∞⋂i=0

3 X \ S ∈ σ for S ∈ σ.We say that (X , σ) is a measurable space, and S ∈ σ aremeasurable sets.

By de Morgan we have also:∞⋃

Measures and Measurable Functions

DefinitionGiven a measurable space (X , σ) then µ : σ → R+ is a (finite)measure if

1 µ(∅) = 0 (for µ(X ) = 1 we have a probability measure).

2 µ(∞⋃

Si) =∞∑

µ(Si) for Si ∈ σ with Si ∩ Sj = ∅ for i 6= j .

DefinitionA function T : X → X ′ between two measure spaces spaces(X , σ, µ) and (X ′, τ ′, µ′) is calledmeasurable iff

∀S ∈ σ′ : T−1(S) ∈ σ.

measure preserving iff ∀S ∈ σ′ also µ′(S′) = µ(T−1(S)).

Topological Mixing Notions

DefinitionGiven a topological dynamical system (G, π,X ).We say that (G, π,X ) is topologically transitive if

∃x ∈ X : OG(x) is dense in X .

We say that (G, π,X ) is (topologically) minimal if

∀x ∈ X : OG(x) is dense in X .

DefinitionA discrete topological dynamical system (T,X ) is calledtopologically (strong) mixing if

∀U,V ⊆ X open and non-empty ∃N : ∀n > N : Tn(U) ∩ V 6= ∅.

Topological Mixing Notions

DefinitionGiven a topological dynamical system (G, π,X ).We say that (G, π,X ) is topologically transitive if

∃x ∈ X : OG(x) is dense in X .

We say that (G, π,X ) is (topologically) minimal if

∀x ∈ X : OG(x) is dense in X .

DefinitionA discrete topological dynamical system (T,X ) is calledtopologically (strong) mixing if

∀U,V ⊆ X open and non-empty ∃N : ∀n > N : Tn(U) ∩ V 6= ∅.

Topologically Transitive

TheoremGiven a discrete topological dynamical system (T,X ) on acompact metric space X then the following conditions areequivalent:

1 ∀x ∈ X : OT(x) is dense in X (topologically transitive).2 ∀C ⊆ X closed with T(C) = C ⇒ C = X or C = ∅.3 ∀O ⊆ X open with T(O) = O ⇒ O = X or O = ∅.

4 ∀O ⊆ X open and non-empty, then∞⋃

n=−∞Tn(O) = X.

Measure Theoretic Mixing Notions

DefinitionGiven a discrete measure theoretic dynamical system (T,X ).We say (T,X ) is measure theoretic transitive or ergodic if

∀S ⊆ X measurable with T(S) = S ⇒ µ(S) = 0 or µ(S) = 1.

DefinitionA discrete measure theoretic dynamical system (T,X ) is calledstrong mixing if

∀S1,S2 ⊆ X measurable limn→∞

µ(T−n(S1) ∩ S2) = µ(S1)µ(S2).

Measure Theoretic Mixing Notions

DefinitionGiven a discrete measure theoretic dynamical system (T,X ).We say (T,X ) is measure theoretic transitive or ergodic if

∀S ⊆ X measurable with T(S) = S ⇒ µ(S) = 0 or µ(S) = 1.

DefinitionA discrete measure theoretic dynamical system (T,X ) is calledstrong mixing if

∀S1,S2 ⊆ X measurable limn→∞

µ(T−n(S1) ∩ S2) = µ(S1)µ(S2).

Ergodicity

TheoremGiven a discrete measure theoretic dynamical system (T,X )with T measure preserving. Then the following conditions areequivalent (with ergodic):

1 ∀S ⊆ X measurable T(S) = S ⇒ µ(S) = 0 or µ(S) = 1.2 ∀S ⊆ X measurable and µ(T−1(S)

aS) = 0

⇒ µ(S) = 0 or µ(S) = 1.

3 ∀S ⊆ X measurable and µ(S) > 0⇒ µ(∞⋃

n=−∞T−n(S)) = 1.

4 ∀S1,S2 ⊆ X measurable and µ(S1) > 0 < µ(S2)⇒ ∃n ∈ N such that µ(T−n(S1) ∩ S2) = 0.

Ergodic Theorem

Given a discrete measure theoretic dynamical system (T,X )and a function (i.e. a random variable) f : X → R.

The phase average of f is defined as µ(f ) =∫

X f (x)dx . The

time average of f is defined as f ∗(x) = limT→∞

0f (Tt (x)))dt .

Theorem (Birkhoff)

Given a discrete measure theoretic dynamical system (T,X ),with T measure preserving, and a function f : X → R withf ∈ L1(X , µ) then the following holds:

(T,X ) is ergodic ⇔ µ(f ) = f ∗(x) µ-almost everywhere.

Ergodic Theorem

Given a discrete measure theoretic dynamical system (T,X )and a function (i.e. a random variable) f : X → R.

The phase average of f is defined as µ(f ) =∫

X f (x)dx . The

time average of f is defined as f ∗(x) = limT→∞

0f (Tt (x)))dt .

Theorem (Birkhoff)

Given a discrete measure theoretic dynamical system (T,X ),with T measure preserving, and a function f : X → R withf ∈ L1(X , µ) then the following holds:

(T,X ) is ergodic ⇔ µ(f ) = f ∗(x) µ-almost everywhere.

Elements of Linear DynamicalSystems

Linear Dynamical System

DefinitionA linear dynamical system is a dynamical system (G, π,X ) with

G is a group (typically G = Z),X is a vector space

and π fulfils theLinearity Axiom:

π(g, .) : X → X is linear ∀g ∈ G.

Many versions of linear dynamical systems play an importantrole in control theory investigating e.g. feed back loops etc.

Linear Dynamical System

DefinitionA linear dynamical system is a dynamical system (G, π,X ) with

G is a group (typically G = Z),X is a vector space

and π fulfils theLinearity Axiom:

π(g, .) : X → X is linear ∀g ∈ G.

Many versions of linear dynamical systems play an importantrole in control theory investigating e.g. feed back loops etc.

Abstract Vector Spaces

DefinitionA Vector Space (over a field K, e.g. R or C) is a set V togetherwith two operations:

Scalar Multiplication . ·. : K× V 7→ VVector Addition .+. : V × V 7→ V

such that (∀x , y , z ∈ V and α, β ∈ K):

1 x + (y + z) = (x + y) + z2 x + y = y + x3 ∃o : x + o = x4 ∃−x : x + (−x) = o

1 α(x + y) = αx + αy2 (α + β)x = αx + βx3 (αβ)x = α(βx)

4 1x = x (1 ∈ K)

Tuple Spaces

TheoremAll finite dimensional vector spaces are isomorphic to the (finite)Cartesian product of the underlying field Kn (i.e. Rn or Cm).

Finite dimensional vectors can always be represented via theircoordinates with respect to a given base, e.g.

x = (x1, x2, x3, . . . , xn)

y = (y1, y2, y3, . . . , yn)

Algebraic Structure

αx = (αx1, αx2, αx3, . . . , αxn)

x + y = (x1 + y1, x2 + y2, x3 + y3, . . . , xn + yn)

Linear Operators

DefinitionA map T : V → W between two vector spaces V andW iscalled a linear map iff

1 T(x + y) = T(x) + T(y) and2 T(αx) = αT(x)

for all x , y ∈ V and all α ∈ K (e.g. K = C or R).

The set of all linear maps between V andW is denotedL(V,W). For V =W we talk about a linear operator on V.

On normed vector spaces the continuous or equivalentlybounded linear operators are of particular interest, i.e.

B(V) = T | ‖T‖ = supx∈V

‖T(x)‖‖x‖

<∞ ⊆ L(V) = L(V,V).

Linear Operators

B(V) = T | ‖T‖ = supx∈V

‖T(x)‖‖x‖

<∞ ⊆ L(V) = L(V,V).

Linear Operators

B(V) = T | ‖T‖ = supx∈V

‖T(x)‖‖x‖

<∞ ⊆ L(V) = L(V,V).

Normed Spaces

DefinitionA complex vector space V is called a normed (vector) space ifthere is a real valued function ‖.‖ on V that satisfies (∀x , y ∈ Vand ∀α ∈ C):

1 ‖x‖ ≥ 02 ‖x‖ = 0 ⇐⇒ x = o3 ‖αx‖ = |α| ‖x‖4 ‖x + y‖ ≤ ‖x‖+ ‖y‖

The function ‖.‖ is called a norm on V.

We have a Banach space if the topology induced by d(x , y)= ‖x − y‖ is complete – always for finite dimensional spaces.

Normed Spaces

DefinitionA complex vector space V is called a normed (vector) space ifthere is a real valued function ‖.‖ on V that satisfies (∀x , y ∈ Vand ∀α ∈ C):

1 ‖x‖ ≥ 02 ‖x‖ = 0 ⇐⇒ x = o3 ‖αx‖ = |α| ‖x‖4 ‖x + y‖ ≤ ‖x‖+ ‖y‖

The function ‖.‖ is called a norm on V.

We have a Banach space if the topology induced by d(x , y)= ‖x − y‖ is complete – always for finite dimensional spaces.

Hilbert Spaces

DefinitionA complex vector space H is called an inner product space (or(pre-)Hilbert space) if there is a complex valued function 〈., .〉on H×H that satisfies (∀x , y , z ∈ H and ∀α ∈ C):

1 〈x , x〉 ≥ 02 〈x , x〉 = 0 ⇐⇒ x = o3 〈αx , y〉 = α 〈x , y〉4 〈x , y + z〉 = 〈x , y〉+ 〈x , z〉5 〈x , y〉 = 〈y , x〉

The function 〈., .〉 is called an inner product on H.

If the topology induced by ‖x‖ =√〈x , x〉 is complete then we

have a Hilbert space – always for finite dimensional spaces.

Basis Vectors

A set of vectors xi is said to be linearly independent iff

λixi =∑

λixi = 0 implies that ∀ i : λi = 0

Two vectors in a Hilbert space are orthogonal iff 〈x , y〉 = 0An orthonormal system (base if it generates all H) in a Hilbertspace is a set of linearly independent vectors bii with:⟨

bi ,bj⟩

= δij =

1 iff i = j0 iff i 6= j

TheoremFor a Hilbert space there exists an orthonormal basis bi. Therepresentation of each vector is unique:

x =∑

xibi =∑

〈x ,bi〉bi

Basis Vectors

λixi =∑

bi ,bj⟩

= δij =

x =∑

xibi =∑

〈x ,bi〉bi

Basis Vectors

λixi =∑

bi ,bj⟩

= δij =

x =∑

xibi =∑

〈x ,bi〉bi

Basis Vectors

λixi =∑

bi ,bj⟩

= δij =

x =∑

xibi =∑

〈x ,bi〉bi

Dual Spaces

A linear functional on a vector space V is a map f : V → K suchthat f (x + y) = f (x) + f (y) and f (αx) = αf (x) for allx , y ∈ V, α ∈ K.

Theorem (Riesz Representation Theorem)Every (bounded) linear functional on a Hilbert space H can berepresented by a vector in the Hilbert space H, such that

f (x) = 〈yf |x〉 = fy (x)

The dual Hilbert space H∗ is isomorphic to the original Hilbertspace H, e.g. for the universal Hilbert space `2(N)∗ = `2(N).

`p(X) =

(xi)i∈X |

(∑i∈X|xi |2

Dual Spaces

f (x) = 〈yf |x〉 = fy (x)

`p(X) =

(xi)i∈X |

(∑i∈X|xi |2

Dual Spaces

f (x) = 〈yf |x〉 = fy (x)

`p(X) =

(xi)i∈X |

(∑i∈X|xi |2

Finite-Dimensional Hilbert Spaces

We represent vectors and their transpose using coordinates:

, ~y =

= (y1, . . . , yn)

The adjoint of ~x = (x1, . . . , xn), with .∗ = . denoting complexconjugate in C), is given by

~x† = ~x∗ = (x∗1 , . . . , x∗n )T

The inner product is:⟨~y , ~x

⟩=∑

y∗i xi = ~y†~x

, ~y =

= (y1, . . . , yn)

~x† = ~x∗ = (x∗1 , . . . , x∗n )T

⟩=∑

y∗i xi = ~y†~x

, ~y =

= (y1, . . . , yn)

~x† = ~x∗ = (x∗1 , . . . , x∗n )T

⟩=∑

y∗i xi = ~y†~x

Differential Equations

Discrete (Time) Dynamical Systems: Collatz

The Colltaz problem is a (one-sided) discrete time dynamicalsystem (C,Z), which we can describe by the followingtransformation:

C : Z→ Z

C(n) =

n/2 if n is even

3× n + 1 otherwise.

The unsolved question is:

Does ∃m ∈ N such that Cm(n) = 1 for all n ∈ N?

Discrete (Time) Dynamical Systems: Collatz

The Colltaz problem is a (one-sided) discrete time dynamicalsystem (C,Z), which we can describe by the followingtransformation:

C : Z→ Z

C(n) =

n/2 if n is even

3× n + 1 otherwise.

The unsolved question is:

Does ∃m ∈ N such that Cm(n) = 1 for all n ∈ N?

Continuous Dynamical Systems

A popular way to specify continuous time dynamical systems isvia (ordinary) differential equations, e.g. Morris W. Hirsch,Stephen Smale, and Robert L. Devaney. Differential Equations,Dynamical Systems and An Introduction to Chaos. Elsevier,2004.

The group action is interpreted as time t ∈ R.

Ordinary Differential Equations

x ′1 = dx1dt = f1(t , x1, x2, . . . , xn)

x ′2 = dx2dt = f2(t , x1, x2, . . . , xn)

. . . . . . . . .

x ′n = dxndt = fn(t , x1, x2, . . . , xn)

Continuous Dynamical Systems

A popular way to specify continuous time dynamical systems isvia (ordinary) differential equations, e.g. Morris W. Hirsch,Stephen Smale, and Robert L. Devaney. Differential Equations,Dynamical Systems and An Introduction to Chaos. Elsevier,2004.

The group action is interpreted as time t ∈ R.

Ordinary Differential Equations

x ′1 = dx1dt = f1(t , x1, x2, . . . , xn)

x ′2 = dx2dt = f2(t , x1, x2, . . . , xn)

. . . . . . . . .

x ′n = dxndt = fn(t , x1, x2, . . . , xn)

Differential

Given a function f : R→ R. We say it is differentiable at a pointt ∈ R if there is a linear map Df (t) : R→ R which approximatesf at t . That is, ∀ε > 0 there is a neigborhood U of t such that:

‖f (t ′)− f (t)− Df (t)(t − t ′)‖ < ε‖t − t ′‖ ∀t ′ ∈ U

We also write for the differential (quotient) Df = dfdt .

We also approximate a function f : Rn → Rm by a linear mapDf (x) : Rn → Rm represented by the matrix of partial derivates:

(Df )ij =∂fi∂tj

i = 1, . . . ,m, i = 1, . . . ,n.

Differential

Given a function f : R→ R. We say it is differentiable at a pointt ∈ R if there is a linear map Df (t) : R→ R which approximatesf at t . That is, ∀ε > 0 there is a neigborhood U of t such that:

‖f (t ′)− f (t)− Df (t)(t − t ′)‖ < ε‖t − t ′‖ ∀t ′ ∈ U

We also write for the differential (quotient) Df = dfdt .

We also approximate a function f : Rn → Rm by a linear mapDf (x) : Rn → Rm represented by the matrix of partial derivates:

(Df )ij =∂fi∂tj

i = 1, . . . ,m, i = 1, . . . ,n.

Euler’s Number

What is Euler’s e? Metafont users know, it is e = 2.7183 . . .It is the unique number such that for f (t) = et we have

(t) = f (t)

The exponential function is the fixed-point or eigen-function ofthe differential operator d

dt . One could show this via the Taylorexpansion of exp(x) =

∑∞n=0

n! as ddt

n! = ntn−1

n(n−1)! = tn−1

(n−1)! .

The simplest differential equation one can think of is perhaps:

x ′(t) =dxdt

(t) = ax(t)

The solution is x(t) = keat = k exp(at) for some constant k(can be determined via an initial/boundary condition, e.g. x(0)).

Euler’s Number

(t) = f (t)

∑∞n=0

n! as ddt

n! = ntn−1

n(n−1)! = tn−1

(n−1)! .

x ′(t) =dxdt

(t) = ax(t)

Euler’s Number

(t) = f (t)

∑∞n=0

n! as ddt

n! = ntn−1

n(n−1)! = tn−1

(n−1)! .

x ′(t) =dxdt

(t) = ax(t)

Euler’s Number

(t) = f (t)

∑∞n=0

n! as ddt

n! = ntn−1

n(n−1)! = tn−1

(n−1)! .

x ′(t) =dxdt

(t) = ax(t)

Ordinary Linear Differential Equations [1, p129]

Solution to ordinary differential equations via exponentation.

x ′1 = dx1dt = a11x1 + a12x2 + . . .+ a1nxn

x ′2 = dx2dt = a21x1 + a22x2 + . . .+ a2nxn

. . . . . . . . .

x ′n = dx2dt = an1x1 + an2x2 + . . .+ annxn

TheoremLet A be an n × n matrix. Then the unique solution to the initialvalue problem x′ = Ax with x(0) = x0 is given by

x(t) = exp(tA)x0.

Ordinary Linear Differential Equations [1, p129]

Solution to ordinary differential equations via exponentation.

x ′1 = dx1dt = a11x1 + a12x2 + . . .+ a1nxn

x ′2 = dx2dt = a21x1 + a22x2 + . . .+ a2nxn

. . . . . . . . .

x ′n = dx2dt = an1x1 + an2x2 + . . .+ annxn

TheoremLet A be an n × n matrix. Then the unique solution to the initialvalue problem x′ = Ax with x(0) = x0 is given by

x(t) = exp(tA)x0.

Computing Solutions

The exponential of a matrix A can be computed as:

exp(A) =∞∑

However this anything but an efficient way to compute it.

We can represent matrices e.g. in Jordan normal form:A = D + N where D is a diagonal matrix and N is an upperdiagonal matrix which is nilpotent, i.e. ∃m s.t. Nm vanishes.This boils down to finding the eigenvalues of A (via SVD).

We then have exp(A) = exp(D + N) = exp(D) exp(N) withexp(diag(d1, . . . ,dn)) = diag(exp(d1), . . . ,exp(dn)) and Nk 6= 0only for finitely many terms.

Computing Solutions

exp(A) =∞∑

Computing Solutions

exp(A) =∞∑

Computing Solutions

exp(A) =∞∑

Smooth Functions

We say a function f is differentiable on R or U ∈ R if it isdifferentiable at every point t ∈ R or t ∈ U. We then writef ∈ C1(R) = C1 or f ∈ C(U).

We can see Df (x) itself as a function Rn → L(Rn,Rm) = Rnm.

As such we can ask if this is itself differentiable. We denote theset of p-times differentiable maps by Cp and by C∞ the set ofinfinitely differentiable or smooth functions.

Note: Differentiation is primarily a real number notion. We needto introduce the notion of a differentiable manifold as a spacewhich looks like Rm locally (with respect to diff. operations).

Smooth Functions

Differentiable Manifolds

DefinitionLet M be a topological space.

A chart (V ,Φ) is a homeomorphism Φ of an open set V of Minto an open set of Rm.

Two charts (V1,Φ1) and (V2,Φ2) are said to be compatible incase V1 ∩ V2 = ∅ or the restricted maps Φ1 Φ−1

2 and Φ2 Φ−11

are in C∞(Rm).

A atlas is a set of compatible charts that cover all of M. Twoatlases are compatible if all their charts are.

A differentiable manifold is a separable, metrizisable space withan set of compatible (equivalent) atlases.

2 and Φ2 Φ−11

are in C∞(Rm).

2 and Φ2 Φ−11

are in C∞(Rm).

2 and Φ2 Φ−11

are in C∞(Rm).

Tangents

DefinitionLet f ,g ∈ L(V,W) then we say that f is tangent to g at t iff

limt ′→t

‖f (t ′)− g(t ′)‖‖t ′ − t‖

Definition

Let M be a manifold and m ∈ M. A curve at m is a C1 mapc : I → M with an open interval in R containing 0 s.t. c(0) = m.

We say that two curves c1 and c2 are tangent if Φ c1 andΦ c2 are tangent at 0.

The tangent space Tm(M) of M at m is the set of (tangent)equivalent classes of curves.

Tangents

limt ′→t

‖f (t ′)− g(t ′)‖‖t ′ − t‖

Definition

Tangents

limt ′→t

‖f (t ′)− g(t ′)‖‖t ′ − t‖

Definition

Lie Groups

DefinitionA Lie group over a field K is a group G equipped with thestructure of a differentiable manifold over K sich that

. · . : G ×G→ G is differentable.

Using the implicit function theorem, one can also show thatg 7→ g−1 is differentiable (a diffeomorphism).

The fields we are typically interested are K = R or K = C.

Lie Groups

Fields

DefinitionA field is a set K together with two operations:

Addition .+. : K×K 7→ KMultiplication . ·. : K×K 7→ K

such that

1 ∀x , y , z ∈ K : x + (y + z) = (x + y) + z2 ∃ o ∈ K, ∀x ∈ K : o + x = x3 ∀ x ∈ K, ∃ −x ∈ K : x + (−x) = o4 ∀x , y ∈ K : x + y = x + y5 ∀x , y , z ∈ K : x · (y · z) = (x · y) · z6 ∃ o 6= e ∈ K, ∀x ∈ K : e · x = x7 ∀ o 6= x ∈ K, ∃x−1 ∈ K : x · x−1 = e8 ∀x , y ∈ K : x · y = y · x9 x · (y + z) = x · y + x · z, ∀x , y , z ∈ K

Examples of Lie Groups

Examples of Lie groups we can mentions here:

The additive group of the field K = K+.The multiplicative group of the field K×.The “circle” T = z ∈ C | |z| = 1 or eiφ | φ ∈ [0,2π).GLn(K) of invertible matrices of order n over K.SLn(K) of matrices of order n over K with det = 1.On(K) orthogonal matrices over K of order n.Un unitary matrices over C of order n.SOn(K) = On(K) ∩ SLn(K).SUn = Un ∩ SLn(C).

Lie Algebras

DefinitionA Lie algebra is a vector space g over some field K togetherwith a binary operation [·, ·] : g× g→ g called the Lie bracket,which satisfies the following:

Bilinearity: ∀α, β ∈ K and ∀x , y , z ∈ g

[αx + βy , z] = α[x , z] + β[y , z]

[z, αx + βy ] = α[z, x ] + β[z, y ]

Alternating on g: ∀x ∈ g[x , x ] = 0

Jacobi identity: ∀x , y , z ∈ g

[x , [y , z]] + [z, [x , y ]] + [y , [z, x ]] = 0

Examples of Lie Algebras

It follows easily ∀x , y ∈ g that [x , y ] = −[y , x ]. One could alsodefine a associative product on an algebra g and then introducethe Lie bracket as [x , y ] = xy − yx .

TheoremGiven a Lie group G then the tangent space at the unit g = TeGis a Lie algebra.

Let g(t) and h(t) be differentiable paths or C1 curves on G.Assume, g(0) = h(0) = e as well as dg

dt (0) = ξ and dhdt (0) = η

then we define a Lie bracket on the tangent space Te(G) via

[ξ, η] =∂2

∂t∂s[g(t),h(s)]|s=t=0

where [g,h] = ghg−1h−1 is the group commutator.

[ξ, η] =∂2

∂t∂s[g(t),h(s)]|s=t=0

[ξ, η] =∂2

∂t∂s[g(t),h(s)]|s=t=0

Prescribed Velocities

DefinitionA path or curve g(t) in a Lie group G with t ∈ R is called aone-parameter subgroup if

g(t + s) = g(t)g(s).

We denote by gξ(s) the one-parameter sub-group withg′ = dg

dt (s) = ξ(s) – i.e. with prescribed “velocity” ξ(s).

DefinitionFor a Lie group G and ξ ∈ g, i.e. its Lie algebra, we define:

exp(ξ) = gξ(1)

g(t + s) = g(t)g(s).

exp(ξ) = gξ(1)

g(t + s) = g(t)g(s).

exp(ξ) = gξ(1)

Exponentation

TheoremThe exponential map exp : g→ G maps a neighbourhood ofzero in the tangent algebra g = Te(G) diffeomorphically onto aneighbourhood of the identity in G.

TheoremLet g = a1 ⊕ . . .⊕ ak be a decomposition of a Lie algebra asdirect sum, then ξ1 + . . .+ ξk 7→ exp(ξ1) . . . exp(ξk ) maps aneighbourhood of zero in g diffeomorphically onto aneighbourhood of the identity in G.

If G is the group of invertible elements in an associative algebra(e.g. of non-singular matrices), then

exp(ξ) =∞∑

Exponentation

exp(ξ) =∞∑

Exponentation

exp(ξ) =∞∑

Stochastic Dynamics

Discrete Time Markov Chains (DTCM)

DefinitionA discrete time Markov chain (DTMC) on S is defined via astochastic matrix P, i.e. an r × r (square) matrix with entries0 ≤ pij ≤ 1 and such that all row sums are equal to one, i.e.∑

pij = 1.

This defines a discrete linear dynamical system:

Phase group: Z or N,Phase space: Rr ,Group action: π(n, x) = x · Pn.

Discrete Time Markov Chains (DTCM)

DefinitionA discrete time Markov chain (DTMC) on S is defined via astochastic matrix P, i.e. an r × r (square) matrix with entries0 ≤ pij ≤ 1 and such that all row sums are equal to one, i.e.∑

pij = 1.

This defines a discrete linear dynamical system:

Phase group: Z or N,Phase space: Rr ,Group action: π(n, x) = x · Pn.

Memoryless Property of DTMC

Let I be a finite (or maybe countable) set. Each i ∈ I is called astate or index. Given a probability space (Ω, σ,P) a randomvariable is a map X : Ω→ I.

A sequence of random variables Xn is a Markov Chain if

P(Xn+1 = i + 1 | X0 = i0,X1 = i1, . . . ,Xn = in) =

= P(Xn+1 = i + 1 | Xn = in) =

= pin,in+1P(Xn = in)

The probability P(i →n j) of reaching state (actually index) jfrom i in exactly n steps is given by p(n)

ij i.e. the entry in row iand column j of Pn.

P(Xn+1 = i + 1 | X0 = i0,X1 = i1, . . . ,Xn = in) =

= P(Xn+1 = i + 1 | Xn = in) =

P(Xn+1 = i + 1 | X0 = i0,X1 = i1, . . . ,Xn = in) =

= P(Xn+1 = i + 1 | Xn = in) =

Properties

DefinitionGiven a DTMC with transition matrix P. A state i is said to be

recurrent if P(i →n i for infinitely many n = 1transient if P(i →n i for infinitely many n = 0

DefinitionA DTMC with transition matrix P is called

ergodic or irreducible: if ∀i , j ∃n such that Pnij > 0.

regular: if ∃n such that ∀i , j we have Pnij > 0.

Properties

DefinitionGiven a DTMC with transition matrix P. A state i is said to be

recurrent if P(i →n i for infinitely many n = 1transient if P(i →n i for infinitely many n = 0

DefinitionA DTMC with transition matrix P is called

ergodic or irreducible: if ∀i , j ∃n such that Pnij > 0.

regular: if ∃n such that ∀i , j we have Pnij > 0.

Long Run Behaviour

TheoremGiven a DTMC with transition matrix P. If it is regular and v anarbitrary probability vector. Then

limn→∞

vPn = w

where w is the unique probability vector for P.

TheoremGiven a DTMC with transition matrix P. Assume P is ergodic.Let An be the matrix defined by:

An =I + P + . . .+ Pn

then An →W where W is a matrix all of whose rows are equalto the unique vector w for P.

Long Run Behaviour

TheoremGiven a DTMC with transition matrix P. If it is regular and v anarbitrary probability vector. Then

limn→∞

vPn = w

where w is the unique probability vector for P.

TheoremGiven a DTMC with transition matrix P. Assume P is ergodic.Let An be the matrix defined by:

An =I + P + . . .+ Pn

then An →W where W is a matrix all of whose rows are equalto the unique vector w for P.

Continuous Time Markov Chains (CTMC)

DefinitionA continuous time Markov chain (CTMC) on S = s1, . . . , sr isdefined via an r × r (square) generator or Q-matrix Q = (qij)specifying the rates going from an index or state i to an index orstate j and which fullfills:

1 0 ≤ −qii <∞ for all i2 qij ≥ 0 for all i 6= j

qij = 0 for all i

Continuous Time Markov Chains (CTMC)

DefinitionA continuous time Markov chain (CTMC) on S = s1, . . . , sr isdefined via an r × r (square) generator or Q-matrix Q = (qij)specifying the rates going from an index or state i to an index orstate j and which fullfills:

1 0 ≤ −qii <∞ for all i2 qij ≥ 0 for all i 6= j

qij = 0 for all i

Computing Transition Probabilities

Again we use exponentation to get the transition probabilities.

P(t) = exp(tQ) =∞∑

This gives the unique solutions to the forward equations

P(t) = P(t)Q with P(0) = I

and the backward equation

P(t) = QP(t) with P(0) = I

and fulfills the (semi-)group property:

P(s + t) = P(s)P(t).

Computing Transition Probabilities

Again we use exponentation to get the transition probabilities.

P(t) = exp(tQ) =∞∑

This gives the unique solutions to the forward equations

P(t) = P(t)Q with P(0) = I

and the backward equation

P(t) = QP(t) with P(0) = I

and fulfills the (semi-)group property:

P(s + t) = P(s)P(t).

Dynamical Systems in Physics

Classical Mechanics (in 10min)

Consider point particles (no volume) with mass m. The positionof a particle is given in some coordinates qi .

The velocity of the particle is given by

vi =dqi

dt= qi

its acceleration is given by

ai =dvdt

dt2 = qi

its momentum is defined as

pi = mqi = mdqi

vi =dqi

dt= qi

ai =dvdt

dt2 = qi

pi = mqi = mdqi

vi =dqi

dt= qi

ai =dvdt

dt2 = qi

pi = mqi = mdqi

vi =dqi

dt= qi

ai =dvdt

dt2 = qi

pi = mqi = mdqi

Lagrange Formalism

Describe the dynamics of a mechanical system via theLagrange function or Lagrangian

L(q1,q2, . . . ,qs, q1, q2, . . . , qs, t)

the action is defined as S =∫ t2

t1L(qi , qi , t)dt .

The Principle of Least Action then implies the Lagrangeequations which give the dynamics:

ddt∂L∂qi− ∂L∂qi

L.D. Landau and E.M. Lifschitz. Mechanik. Akademie-Verlag,Berlin, 1981.

Lagrange Formalism

Describe the dynamics of a mechanical system via theLagrange function or Lagrangian

L(q1,q2, . . . ,qs, q1, q2, . . . , qs, t)

the action is defined as S =∫ t2

t1L(qi , qi , t)dt .

The Principle of Least Action then implies the Lagrangeequations which give the dynamics:

ddt∂L∂qi− ∂L∂qi

Lagrange Examples

Single Particle:L =

(x2 + y2 + z2)

Pendulum: length l , angle φ, mass m, gravitational constant g

l2φ2 + mgl cos(φ)

Double Pendulum: angles φ1 and φ2, lengths l1 and l2, massesm1 and m2 [2, p13]:

L =m1 + m2

2l21 φ

2l22 φ

m2l1l2φ1φ2 cos(φ1 − φ2) +

(m1 + m2)gl1 cos(φ1) + m2gl2 cos(φ2)

Lagrange Examples

Single Particle:L =

(x2 + y2 + z2)

l2φ2 + mgl cos(φ)

L =m1 + m2

2l21 φ

2l22 φ

Lagrange Examples

Single Particle:L =

(x2 + y2 + z2)

l2φ2 + mgl cos(φ)

L =m1 + m2

2l21 φ

2l22 φ

Hamiltonian Formalism

Describe the dynamics of a mechanical system via theHamilton function or Hamiltonian:

H(pi ,qi , t) =∑

pi qi − L(qi , qi , t)

The dynamics of the system is then described via theHamiltonian or canonical equations:

qi =dqdt

=∂H∂pi

qi =dqdt

= −∂H∂qi

Hamiltonian Formalism

Describe the dynamics of a mechanical system via theHamilton function or Hamiltonian:

H(pi ,qi , t) =∑

pi qi − L(qi , qi , t)

The dynamics of the system is then described via theHamiltonian or canonical equations:

qi =dqdt

=∂H∂pi

qi =dqdt

= −∂H∂qi

Hamiltonian Examples

Single Particle:

x + p2y + p2

Particle in Field:

x + p2y + p2

z ) + U(x , y , z)

Pendulum: with pφ = ml2φ and φ =pφ

H =p2φ

2ml2−mgl cos(φ)

Single Particle:

x + p2y + p2

Particle in Field:

x + p2y + p2

z ) + U(x , y , z)

H =p2φ

2ml2−mgl cos(φ)

Single Particle:

x + p2y + p2

Particle in Field:

x + p2y + p2

z ) + U(x , y , z)

H =p2φ

2ml2−mgl cos(φ)

Quantum Mechanics (in 20min)

Arguably, physics is ultimately about explaining experimentsand forecasting measurement results.

Observables: Entities which are (actually) measured when anexperiment is conducted on a system.

State: Entities which completely describe (or model) thesystem we are interested in.

Measurement establishes a relation between states andobservables of a given system. Dynamics describes howobservables and/or the state changes over time.

Related Questions: What is our knowledge of what? How dowe obtain this information? What is a description on how thesystem changes?

Postulates for Quantum Mechanics (ca 1950)

The quantum state of a (free) particle is described by a(normalised) complex valued function:

~ψ ∈ L2(x) i.e.∫|ψ(x)|2dx = 1

Two quantum states can be superimposed, i.e.

α1 ~ψ1 + α2 ~ψ2

Any observable A is represented by a linear, self-adjointoperator A on L2(x).Possible measurement results: Eigenvalues of A,representing the observable A:

A~φi = λi ~φi

Probability to measure λn if the system is in state~ψ =

∑i ψi ~φi is

Pr(A = λn, ~ψ) = ‖~ψn‖2

~ψ ∈ L2(x) i.e.∫|ψ(x)|2dx = 1

α1 ~ψ1 + α2 ~ψ2

A~φi = λi ~φi

∑i ψi ~φi is

Pr(A = λn, ~ψ) = ‖~ψn‖2

~ψ ∈ L2(x) i.e.∫|ψ(x)|2dx = 1

α1 ~ψ1 + α2 ~ψ2

A~φi = λi ~φi

∑i ψi ~φi is

Pr(A = λn, ~ψ) = ‖~ψn‖2

~ψ ∈ L2(x) i.e.∫|ψ(x)|2dx = 1

α1 ~ψ1 + α2 ~ψ2

A~φi = λi ~φi

∑i ψi ~φi is

Pr(A = λn, ~ψ) = ‖~ψn‖2

~ψ ∈ L2(x) i.e.∫|ψ(x)|2dx = 1

α1 ~ψ1 + α2 ~ψ2

A~φi = λi ~φi

∑i ψi ~φi is

Pr(A = λn, ~ψ) = ‖~ψn‖2

Postulates for Quantum Mechanics

Observables and states of a system are represented byhermitian (i.e. self-adjoint) elements a of a C*-algebra A and bystates w (i.e. normalised linear functionals) over this algebra.

Possible results of measurements of an observable a aregiven by the spectrum Sp(a) of an observable. Their probabilitydistribution in a certain state w is given by the probabilitymeasure µ(w) induced by the state w on Sp(a).

Walter Thirring: Quantum Mathematical Physics, 2nd ed.Springer Verlag, 2002

Quantum Mathematics

Quantum physics is often/sometimes counter-intuitive.

However, the standard mathematical model of (closed)quantum systems is relatively simple and just requires somebasic (complex) linear algebra.

The information describing the state of an (isolated)quantum mechanical system is representedmathematically by a (normalised) vector in a complexvector Hilbert space H.An observable are represented mathematically by aself-adjoint matrix (operator) A acting on H.

Two states can be combined to form a new state α |x〉+ β |y〉as long as |α|2 + |β|2 = 1 (Superposition).

Quantum Mathematics

Quantum States and Notation

The state of a QM system is usually denoted by |x〉 ∈ H. Theinner product 〈x |y〉 of two vectors in H – which is describing theangle between them – is very important in QM.

P.A.M. Dirac “invented” the Bra-Ket Notation based on thefollowing simple facts:

Typewriters had no sub-scripts ~xiHilbert spaces have inner product

Simply “take inner product appart” to denote vectors in H:⟨xi , yj

⟩=⟨xi |yj

⟩= 〈i | |j〉

⟩=⟨xi |yj

⟩= 〈i | |j〉

⟩=⟨xi |yj

⟩= 〈i | |j〉

⟩=⟨xi |yj

⟩= 〈i | |j〉

Conventions

Physical Convention:

〈x |αy〉 = α 〈x |y〉

Mathematical Convention:

〈αx , y〉 = α 〈x , y〉

Linear in first or second argument? In mathematics we have:

〈x , αy〉 = 〈αy , x〉 = α〈y , x〉 = α 〈x , y〉

Conventions

〈x |αy〉 = α 〈x |y〉

〈αx , y〉 = α 〈x , y〉

〈x , αy〉 = 〈αy , x〉 = α〈y , x〉 = α 〈x , y〉

Conventions

〈x |αy〉 = α 〈x |y〉

〈αx , y〉 = α 〈x , y〉

〈x , αy〉 = 〈αy , x〉 = α〈y , x〉 = α 〈x , y〉

Quantum Measurement

The expected result of measuring A of a system in state|x〉 ∈ H is given by:

〈A〉x = 〈x |A |x〉 = 〈x | |Ax〉

The only possible results are eigenvalues λi of A.The probability of measuring λn in state |x〉 is

Pr(A = λn, x) = 〈x |Pn |x〉

with Pn the (orthogonal) projection (s.t. A =∑

i λiPi )

d(n)∑j=1

|λn, j〉〈λn, j |

Quantum Measurement

〈A〉x = 〈x |A |x〉 = 〈x | |Ax〉

Pr(A = λn, x) = 〈x |Pn |x〉

i λiPi )

d(n)∑j=1

Quantum Measurement

〈A〉x = 〈x |A |x〉 = 〈x | |Ax〉

Pr(A = λn, x) = 〈x |Pn |x〉

i λiPi )

d(n)∑j=1

Quantum Dynamics

The dynamics of a closed system is described by theSchrödinger Equation:

i~d |x〉

dt= H |x〉

for the (self-adjoint) Hamiltonian H.The solution is a unitary operator Ut = exp(itH).

TheoremFor any self-adjoint operator A the operator

exp(iA) = eiA =∞∑

is a unitary operator.

Quantum Dynamics

i~d |x〉

dt= H |x〉

Quantum Dynamics

i~d |x〉

dt= H |x〉

Adjoint Operator

For a matrix A = (Aij) its transpose matrix AT is defined as

(ATij ) = (Aji)

the conjugate matrix A∗ is defined by

(A∗ij) = (Aij)∗

and the adjoint matrix A† is given via

(A†ij) = (A∗ji) or A† = (A∗)T

Notation: In mathematics the adjoint operator is usuallydenoted by A∗ and defined implicitly via:

〈A(x), y〉 = 〈x ,A∗(y)〉 or 〈A†x |y〉 = 〈x ,Ay〉

Adjoint Operator

(ATij ) = (Aji)

(A∗ij) = (Aij)∗

(A†ij) = (A∗ji) or A† = (A∗)T

〈A(x), y〉 = 〈x ,A∗(y)〉 or 〈A†x |y〉 = 〈x ,Ay〉

Adjoint Operator

(ATij ) = (Aji)

(A∗ij) = (Aij)∗

(A†ij) = (A∗ji) or A† = (A∗)T

〈A(x), y〉 = 〈x ,A∗(y)〉 or 〈A†x |y〉 = 〈x ,Ay〉

Adjoint Operator

(ATij ) = (Aji)

(A∗ij) = (Aij)∗

(A†ij) = (A∗ji) or A† = (A∗)T

〈A(x), y〉 = 〈x ,A∗(y)〉 or 〈A†x |y〉 = 〈x ,Ay〉

Unitary Operators

A square matrix/operator U is called unitary if

U†U = I = UU†

That means U’s inverse is U† = U−1. It also implies that U isinvertible and the inverse is easy to compute.

The postulates of Quantum Mechanics require that the timeevolution to a quantum state, e.g. a qubit, are implemented viaa unitary operator (as long as there is no measurement).

The unitary evolution of an (isolated) quantum state/system is amathematical consequence of being a solution of theSchrödinger equation for some Hamiltonian operator H.

Unitary Operators

U†U = I = UU†

Unitary Operators

U†U = I = UU†

Unitary Operators

U†U = I = UU†

Self Adjoint Operators

An operator A is called self-adjoint or hermitian iff

A = A†

The postulates of Quantum Mechanics require that a quantumobservable A is represented by a self-adjoint operator A.

Possible measurement results are eigenvalues λi of Adefined as

A |i〉 = λi |i〉 or A~ai = λi~ai

Probability to observe λk in state |x〉 =∑

i αi |i〉 is

Pr(A = λk , |x〉) = |αk |2

A = A†

i αi |i〉 is

Pr(A = λk , |x〉) = |αk |2

A = A†

i αi |i〉 is

Pr(A = λk , |x〉) = |αk |2

A = A†

i αi |i〉 is

Pr(A = λk , |x〉) = |αk |2

Projections

ProjectionsAn operator P on Cn is called projection (or idempotent) iff

P2 = PP = P

Orthogonal ProjectionAn operator P on Cn is called (orthogonal) projection iff

P2 = P = P†

We say that an (orthogonal) projection P projects onto itsimage space P(Cn), which is always a linear sub-spaces of Cn.

Birkhoff-von Neumann: Projection on a Hilbert space form anortho-lattice which gives rise to non-classical a “quantum logic”.

Projections

P2 = PP = P

P2 = P = P†

Projections

P2 = PP = P

P2 = P = P†

Projections

P2 = PP = P

P2 = P = P†

Spectral Theorem

In the bra-ket notation we can represent a projection onto thesub-space generated by |x〉 by the outer product Px = |x〉〈x |.

TheoremA self-adjoint operator A (on a finite dimensional Hilbert space,e.g. Cn) can be represented uniquely as a linear combination

A =∑

with λi ∈ R and Pi the (orthogonal) projection onto theeigen-space generated by the eigen-vector |i〉, i.e. Pi = |i〉〈i |

In the degenerate case we had to consider: Pi =∑d(n)

∣∣ij⟩⟨ij ∣∣.Herbert Wiklicky Dynamical Systems

Spectral Theorem

A =∑

Spectral Theorem

A =∑

Measurement Process

If we perform a measurement of the observable represented by:

A =∑

λi |i〉〈i |

with eigen-values λi and eigen-vectors |i〉 in a state |x〉 we haveto decompose the state according to the observable, i.e.

|x〉 =∑

Pi |x〉 =∑

|i〉〈i |x〉 =∑

〈i |x〉 |i〉 =∑

αi |i〉

With probability |αi |2 = | 〈i |x〉 |2 two things happenThe measurement instrument will the display λi .The state |x〉 collapses to |i〉.

Measurement Process

A =∑

λi |i〉〈i |

|x〉 =∑

Pi |x〉 =∑

|i〉〈i |x〉 =∑

〈i |x〉 |i〉 =∑

αi |i〉

Measurement Process

A =∑

λi |i〉〈i |

|x〉 =∑

Pi |x〉 =∑

|i〉〈i |x〉 =∑

〈i |x〉 |i〉 =∑

αi |i〉

Measurement Process

A =∑

λi |i〉〈i |

|x〉 =∑

Pi |x〉 =∑

|i〉〈i |x〉 =∑

〈i |x〉 |i〉 =∑

αi |i〉

Spectrum

The set of eigen-values λ1, λ2, . . . of an operator A is calledits spectrum σ(A).

σ(A) = λ | λI− A is not invertible

It is possible that for an eigen-value λi in the equation

A |i〉 = λi |i〉

we may have more than one eigen-vector |i〉, i.e. the dimensionof the eigen-space d(n) > 1. We will not consider thesedegenerate cases here.

Spectrum

The set of eigen-values λ1, λ2, . . . of an operator A is calledits spectrum σ(A).

σ(A) = λ | λI− A is not invertible

It is possible that for an eigen-value λi in the equation

A |i〉 = λi |i〉

we may have more than one eigen-vector |i〉, i.e. the dimensionof the eigen-space d(n) > 1. We will not consider thesedegenerate cases here.

Heisenberg and Schrödinger Picture

Describe the dynamics in terms of observables or states.

In particular if we consider not just pure (isolated) states, i.e.vectors in a Hilbert space, but instead probabilistic states whichare repesented by density matrices.

A density matrix ρ ∈ B(H) is a Hermitian semi-positive definitematrix or operator with trace(ρ) = 1. Note that a given purestate |ψ〉 can also be represented with density matrix |ψ〉〈ψ|.

The quantum dynamics can be described as for A observableand ρ state (as density matrix/operator).

Schrödinger Picture: %t = U(t)%0U∗(t).Heisenberg Picture: At = U∗(t)A0U(t).

Reversibility

Quantum DynamicsFor unitary transformations describing qubit dynamics:

U† = U−1

The quantum dynamics is invertible or reversible

Quantum MeasurementFor projection operators involved in quantum measurement:

P† 6= P−1

The quantum measurement is not reversible. However

P2 = P

The quantum measurement is idempotent.

Reversibility

U† = U−1

P† 6= P−1

P2 = P

Reversibility

U† = U−1

P† 6= P−1

P2 = P

Dynamics of Programs

pWhile – Syntax I

Full programs contain optional variable declarations:

P ::= begin S end| var D begin S end

Declarations are of the form:

r ::= bool| int| c1, . . . , cn | c1 .. cn

D ::= v : r| v : r ; D

with ci (integer) constants and r denoting ranges.

pWhile – Syntax I

Full programs contain optional variable declarations:

P ::= begin S end| var D begin S end

Declarations are of the form:

r ::= bool| int| c1, . . . , cn | c1 .. cn

D ::= v : r| v : r ; D

with ci (integer) constants and r denoting ranges.

pWhile – Syntax II

The syntax of statements S is as follows:

Where the pi are constants, representing choice probabilities.

pWhile – Syntax II

The syntax of statements S is as follows:

S ::= [stop]`

| [skip]`

| [v := a]`

| [v ?= r ]`

| S1; S2| choose` p1 : S1 or p2 : S2 ro| if [b]` then S1 else S2 fi| while [b]` do S od

Where the pi are constants, representing choice probabilities.

Evaluation of Expressions

σ 3 State = Var→ Z ] T

To illustrate approach consider only finite sub-range of Z.Evaluation E of expressions e in state σ:

E(n)σ = nE(v)σ = σ(v)

E(a1 a2)σ = E(a1)σ E(a2)σ

E(true)σ = ttE(false)σ = ffE(not b)σ = ¬E(b)σ

. . . = . . .

pWhile – SOS Semantics I

R0 〈skip, σ〉⇒1〈stop, σ〉

R1 〈stop, σ〉⇒1〈stop, σ〉

R2 〈v:=e, σ〉⇒1〈stop, σ[v 7→ E(e)σ]〉

R3 〈v?=r , σ〉⇒ 1|r|〈stop, σ[v 7→ ri ∈ r ]〉

R41〈S1, σ〉⇒p〈S′1, σ′〉

〈S1; S2, σ〉⇒p〈S′1; S2, σ′〉

R42〈S1, σ〉⇒p〈stop, σ′〉〈S1; S2, σ〉⇒p〈S2, σ

′〉

pWhile – SOS Semantics II

R51 〈choose p1 : S1 or p2 : S2, σ〉⇒p1〈S1, σ〉

R52 〈choose p1 : S1 or p2 : S2, σ〉⇒p2〈S2, σ〉

R61 〈if b then S1 else S2, σ〉⇒1〈S1, σ〉 if E(b)σ = tt

R62 〈if b then S1 else S2, σ〉⇒1〈S2, σ〉 if E(b)σ = ff

R71 〈while b do S, σ〉⇒1〈S; while b do S, σ〉 if E(b)σ = tt

R72 〈while b do S, σ〉⇒1〈stop, σ〉 if E(b)σ = ff

Factorial

varm : 0..2;n : 0..2;

beginm := 1;while (n>1) do

m := m*n;n := n-1;

od;stop; # loopingend

Multi Variable State

The problem we first consider is how to describe distributionsover the cartesian product in order to represent the probabilitiesthat two or more variables have certain values.

As we have D(S) ⊆ V(S) we investigate V(S × S). In order tounderstand the relation between V(S) and V(S × S) and ingeneral V(Sn) we need to consider the tensor product.

Essential for the further treatment is the fact (more later) that

V(S × S) = V(S)⊗ V(S)

Kronecker Product

Given a n ×m matrix A and a k × l matrix B:

a11 . . . a1m...

. . ....

an1 . . . anm

b11 . . . b1l...

. . ....

bk1 . . . bkl

The tensor or Kronecker product A⊗ B is a nk ×ml matrix:

A⊗ B =

a11B . . . a1mB...

. . ....

an1B . . . anmB

Special cases are square matrices (n = m and k = l) andvectors (row n = k = 1, column m = l = 1).

Kronecker Product

a11 . . . a1m...

. . ....

an1 . . . anm

b11 . . . b1l...

. . ....

bk1 . . . bkl

A⊗ B =

a11B . . . a1mB...

. . ....

an1B . . . anmB

Kronecker Product

a11 . . . a1m...

. . ....

an1 . . . anm

b11 . . . b1l...

. . ....

bk1 . . . bkl

A⊗ B =

a11B . . . a1mB...

. . ....

an1B . . . anmB

Tensor Product Properties

The tensor product of n linear operators A1, A2, . . . , An isassociative (but in general not commutative) and has e.g. thefollowing properties: